Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.8%

Time: 12.5s

Alternatives: 12

Speedup: 1.5×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x \cdot \left(x \cdot x\right)\right|\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (* x (* x x)))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs((x * (x * x)));
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs((x * (x * x)));
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs((x * (x * x)))
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = abs(Float64(x * Float64(x * x)))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs((x * (x * x)));
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left|x \cdot \left(x \cdot x\right)\right|\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot x\right)\right|\right)\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot x\right)\right|\right)\right)\right)\right)\right)\right| \]
4. Add Preprocessing

Alternative 2: 99.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.047619047619047616\right) \cdot \frac{\left|x\right|}{\frac{\frac{\sqrt{\pi}}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (fabs
    (*
     x
     (/ (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)))) (sqrt PI))))
   (*
    (* x 0.047619047619047616)
    (/ (fabs x) (/ (/ (sqrt PI) x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(((double) M_PI)))));
	} else {
		tmp = (x * 0.047619047619047616) * (fabs(x) / ((sqrt(((double) M_PI)) / x) / (x * (x * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = Math.abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / Math.sqrt(Math.PI))));
	} else {
		tmp = (x * 0.047619047619047616) * (Math.abs(x) / ((Math.sqrt(Math.PI) / x) / (x * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / math.sqrt(math.pi))))
	else:
		tmp = (x * 0.047619047619047616) * (math.fabs(x) / ((math.sqrt(math.pi) / x) / (x * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)))) / sqrt(pi))));
	else
		tmp = Float64(Float64(x * 0.047619047619047616) * Float64(abs(x) / Float64(Float64(sqrt(pi) / x) / Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(pi))));
	else
		tmp = (x * 0.047619047619047616) * (abs(x) / ((sqrt(pi) / x) / (x * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * 0.047619047619047616), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[(N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + {x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      19. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. Simplified 99.9%

      \[\leadsto \left|\frac{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}}{\sqrt{\pi}} \cdot x\right| \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left|x\right|\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right| \cdot \color{blue}{\left|\left|x\right| \cdot \frac{1}{21}\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \left(\frac{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \left(\left(\left|x\right| \cdot x\right) \cdot \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\left(\left|x\right| \cdot x\right), \color{blue}{\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\left(x \cdot \left|x\right|\right), \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left|x\right|\right)\right), \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       6. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       16. PI-lowering-PI.f64 99.9%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left(x \cdot \left|x\right|\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{21} \cdot \left(x \cdot \color{blue}{\left(\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{21} \cdot x\right) \cdot \color{blue}{\left(\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{21} \cdot x\right), \color{blue}{\left(\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \left(\color{blue}{\left|x\right|} \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \left(\left|x\right| \cdot \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}}\right)\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \left(\frac{\left|x\right|}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\left(\left|x\right|\right), \color{blue}{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)}\right)\right) \]

       8. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{x}}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{x}\right), \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), x\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), x\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), x\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), x\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{21}, x\right), \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot \frac{\left|x\right|}{\frac{\frac{\sqrt{\pi}}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.047619047619047616\right) \cdot \frac{\left|x\right|}{\frac{\frac{\sqrt{\pi}}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left(x \cdot \left|x\right|\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (fabs
    (*
     x
     (/ (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)))) (sqrt PI))))
   (*
    0.047619047619047616
    (* (* x (fabs x)) (/ (* x (* x (* x (* x x)))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(((double) M_PI)))));
	} else {
		tmp = 0.047619047619047616 * ((x * fabs(x)) * ((x * (x * (x * (x * x)))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = Math.abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / Math.sqrt(Math.PI))));
	} else {
		tmp = 0.047619047619047616 * ((x * Math.abs(x)) * ((x * (x * (x * (x * x)))) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / math.sqrt(math.pi))))
	else:
		tmp = 0.047619047619047616 * ((x * math.fabs(x)) * ((x * (x * (x * (x * x)))) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)))) / sqrt(pi))));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(x * abs(x)) * Float64(Float64(x * Float64(x * Float64(x * Float64(x * x)))) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(pi))));
	else
		tmp = 0.047619047619047616 * ((x * abs(x)) * ((x * (x * (x * (x * x)))) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.047619047619047616 * N[(N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + {x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      19. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. Simplified 99.9%

      \[\leadsto \left|\frac{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}}{\sqrt{\pi}} \cdot x\right| \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left|x\right|\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left|\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right| \cdot \color{blue}{\left|\left|x\right| \cdot \frac{1}{21}\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \left(\frac{\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \left(\left(\left|x\right| \cdot x\right) \cdot \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\left(\left|x\right| \cdot x\right), \color{blue}{\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\left(x \cdot \left|x\right|\right), \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left|x\right|\right)\right), \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       6. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       16. PI-lowering-PI.f64 99.9%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\left(\left(x \cdot \left|x\right|\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left(x \cdot \left|x\right|\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (fabs (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* x (* x 0.6666666666666666))))))
   (/ (fabs (* x (* (* x x) (* (* x x) 0.2)))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = fabs((x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = fabs((x * ((x * x) * ((x * x) * 0.2)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = Math.abs((x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = Math.abs((x * ((x * x) * ((x * x) * 0.2)))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = math.fabs((x * (math.sqrt((1.0 / math.pi)) * (2.0 + (x * (x * 0.6666666666666666))))))
	else:
		tmp = math.fabs((x * ((x * x) * ((x * x) * 0.2)))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = abs(Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = Float64(abs(Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.2)))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = abs((x * (sqrt((1.0 / pi)) * (2.0 + (x * (x * 0.6666666666666666))))));
	else
		tmp = abs((x * ((x * x) * ((x * x) * 0.2)))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[Abs[N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + 2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}, x\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{2}{3} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(\frac{2}{3} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right)\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right)\right), x\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right), x\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

      13. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} \cdot x\right| \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left|\frac{\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. fabs-div N/A

         \[\leadsto \frac{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}} \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|}{\sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      4. add-sqr-sqrt N/A

         \[\leadsto \frac{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right|}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left|\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. fabs-mul N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left|\left|x\right|\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. fabs-fabs N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left|x\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. fabs-mul N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left|x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(x \cdot \left(\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right|}}{\sqrt{\pi}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2} \cdot {x}^{2}, \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{\left(2 \cdot 2\right)}, \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{4}, \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right) + 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       12. fma-define N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{4}, \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{\left(2 \cdot 2\right)}, \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       14. pow-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2} \cdot {x}^{2}, \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       15. fma-define N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 88.8%

       \[\leadsto \frac{\left|\color{blue}{x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)\right)}\right|}{\sqrt{\pi}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{5}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   14. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{5} \cdot {x}^{\left(4 + 1\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{5} \cdot \left({x}^{4} \cdot x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(\left(\frac{1}{5} \cdot {x}^{4}\right) \cdot x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(\frac{1}{5} \cdot {x}^{4}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{5} \cdot {x}^{4}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       16. *-lowering-*.f64 88.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   15. Simplified 88.8%

       \[\leadsto \frac{\left|\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{0.6666666666666666}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (* (fabs (* x (* x x))) (/ 0.6666666666666666 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((x * (x * x))) * (0.6666666666666666 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((x * (x * x))) * (0.6666666666666666 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((x * (x * x))) * (0.6666666666666666 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(x * Float64(x * x))) * Float64(0.6666666666666666 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((x * (x * x))) * (0.6666666666666666 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 98.9%

      \[\leadsto \left|\frac{\color{blue}{2}}{\sqrt{\pi}} \cdot x\right| \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. Simplified 78.5%

      \[\leadsto \left|\frac{\color{blue}{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}}{\sqrt{\pi}} \cdot x\right| \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       4. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   12. Simplified 78.5%

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot x\right) \cdot 0.6666666666666666}}{\sqrt{\pi}} \cdot x\right| \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|x \cdot \frac{\left(x \cdot x\right) \cdot \frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       2. associate-/l* N/A

          \[\leadsto \left|x \cdot \left(\left(x \cdot x\right) \cdot \frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right| \]

       3. associate-*r* N/A

          \[\leadsto \left|\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       4. fabs-mul N/A

          \[\leadsto \left|x \cdot \left(x \cdot x\right)\right| \cdot \color{blue}{\left|\frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right|} \]

       5. fabs-div N/A

          \[\leadsto \left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{\left|\frac{2}{3}\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}} \]

       6. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{\frac{2}{3}}{\left|\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right|} \]

       7. rem-sqrt-square N/A

          \[\leadsto \left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{\frac{2}{3}}{\sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

       8. add-sqr-sqrt N/A

          \[\leadsto \left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left|x \cdot \left(x \cdot x\right)\right|\right), \color{blue}{\left(\frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

       10. fabs-lowering-fabs.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{\color{blue}{\frac{2}{3}}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{2}{3}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{2}{3}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

       14. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right) \]

       15. PI-lowering-PI.f64 79.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   14. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{\left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{0.6666666666666666}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(x \cdot x\right)\right| \cdot \frac{0.6666666666666666}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{\left(x \cdot x\right) \cdot 0.6666666666666666}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* x (/ (* (* x x) 0.6666666666666666) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((x * (((x * x) * 0.6666666666666666) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((x * (((x * x) * 0.6666666666666666) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((x * (((x * x) * 0.6666666666666666) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(x * Float64(Float64(Float64(x * x) * 0.6666666666666666) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((x * (((x * x) * 0.6666666666666666) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 98.9%

      \[\leadsto \left|\frac{\color{blue}{2}}{\sqrt{\pi}} \cdot x\right| \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

      3. fabs-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

      4. mul-fabs N/A

         \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. Simplified 78.5%

      \[\leadsto \left|\frac{\color{blue}{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}}{\sqrt{\pi}} \cdot x\right| \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       4. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   12. Simplified 78.5%

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot x\right) \cdot 0.6666666666666666}}{\sqrt{\pi}} \cdot x\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{\left(x \cdot x\right) \cdot 0.6666666666666666}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (/
    (+
     2.0
     (*
      (* x x)
      (+
       0.6666666666666666
       (* x (* x (+ 0.2 (* (* x x) 0.047619047619047616)))))))
    (sqrt PI)))))

C

double code(double x) {
	return fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(x * Float64(x * Float64(0.2 + Float64(Float64(x * x) * 0.047619047619047616))))))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(x * N[(x * N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Final simplification 99.9%

   \[\leadsto \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}}\right| \]
8. Add Preprocessing

Alternative 8: 93.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (/ (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)))) (sqrt PI)))))

C

double code(double x) {
	return fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + {x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   8. pow-sqr N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2} + \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   19. *-lowering-*.f64 96.3%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 96.3%

   \[\leadsto \left|\frac{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}}{\sqrt{\pi}} \cdot x\right| \]

10. Final simplification 96.3%

    \[\leadsto \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 9: 89.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	return fabs((x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (x * (x * 0.6666666666666666))))));
}

Java

public static double code(double x) {
	return Math.abs((x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (x * (x * 0.6666666666666666))))));
}

Python

def code(x):
	return math.fabs((x * (math.sqrt((1.0 / math.pi)) * (2.0 + (x * (x * 0.6666666666666666))))))

Julia

function code(x)
	return abs(Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (sqrt((1.0 / pi)) * (2.0 + (x * (x * 0.6666666666666666))))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + 2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}, x\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{2}{3} \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(\frac{2}{3} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right)\right), x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right)\right), x\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right), x\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

   13. *-lowering-*.f64 92.8%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{2}{3}\right)\right)\right)\right), x\right)\right) \]

9. Simplified 92.8%

   \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} \cdot x\right| \]

10. Final simplification 92.8%

    \[\leadsto \left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right| \]
11. Add Preprocessing

Alternative 10: 89.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* x (/ (+ 2.0 (* x (* x 0.6666666666666666))) (sqrt PI)))))

C

double code(double x) {
	return fabs((x * ((2.0 + (x * (x * 0.6666666666666666))) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + (x * (x * 0.6666666666666666))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + (x * (x * 0.6666666666666666))) / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + (x * (x * 0.6666666666666666))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 92.8%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{2}{3}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 92.8%

   \[\leadsto \left|\frac{\color{blue}{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}}{\sqrt{\pi}} \cdot x\right| \]

10. Final simplification 92.8%

    \[\leadsto \left|x \cdot \frac{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 11: 67.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (sqrt (/ 1.0 PI)) (* 2.0 x))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 * x)));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 * x)));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 * x)))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (2.0 * x)));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(2 \cdot \left(x \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(2 \cdot x\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(2 \cdot x\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(2 \cdot x\right)\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(2 \cdot x\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \left(x \cdot 2\right)\right)\right) \]

   8. *-lowering-*.f64 69.5%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(x, 2\right)\right)\right) \]

9. Simplified 69.5%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)}\right| \]

10. Final simplification 69.5%

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x\right)\right| \]
11. Add Preprocessing

Alternative 12: 67.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

double code(double x) {
	return fabs((x * (2.0 / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * (2.0 / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(2.0 / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (2.0 / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]

   2. fabs-mul N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{\left|\left|x\right|\right|} \]

   3. fabs-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \left|x\right| \]

   4. mul-fabs N/A

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right| \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}{\sqrt{\pi}} \cdot x\right|} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

9. Simplified 69.2%

   \[\leadsto \left|\frac{\color{blue}{2}}{\sqrt{\pi}} \cdot x\right| \]

10. Final simplification 69.2%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))